# How do you simplify f(theta)=-2csc(theta/4)-cot(theta/2)-3cos(theta/4) to trigonometric functions of a unit theta?

Feb 10, 2017

$f \left(\theta\right) = \pm \frac{2}{\sqrt{\frac{1 \pm \sqrt{\frac{1 + \cos \left(\theta\right)}{2}}}{2}}} \pm \frac{\sqrt{\frac{1 \pm \sqrt{\frac{1 + \cos \left(\theta\right)}{2}}}{2}}}{\sqrt{\frac{1 - \cos \left(\theta\right)}{2}}} \pm 3 \sqrt{\frac{1 \pm \sqrt{\frac{1 + \cos \left(\theta\right)}{2}}}{2}}$

#### Explanation:

Assuming the question is asking you to get rid of the fractions, it is important to know your half-angle formulas:

• sin(u/2)=+-sqrt((1-cos(u))/2
• $\cos \left(\frac{u}{2}\right) = \pm \sqrt{\frac{1 + \cos \left(u\right)}{2}}$
• $\tan \left(\frac{u}{2}\right) = \frac{1 - \cos \left(u\right)}{\sin} \left(u\right)$

Step 1. Change $\csc$ and $\cot$ in terms of $\sin$ and $\cos$

$f \left(\theta\right) = - \frac{2}{\sin} \left(\frac{\theta}{4}\right) - \cos \frac{\frac{\theta}{2}}{\sin} \left(\frac{\theta}{2}\right) - 3 \cos \left(\frac{\theta}{4}\right)$

Step 2. Express each term so that it has $\theta \text{/} 2$

$f \left(\theta\right) = - \frac{2}{\sin} \left(\frac{\theta \text{/"2)/2)-cos(theta/2)/sin(theta/2)-3cos((theta"/} 2}{2}\right)$

Step 3. Replace each $\sin$ and $\cos$ with half-angle formulas

$f \left(\theta\right) = \pm \frac{2}{\sqrt{\frac{1 - \cos \left(\frac{\theta}{2}\right)}{2}}} \pm \frac{\sqrt{\frac{1 + \cos \left(\frac{\theta}{2}\right)}{2}}}{\sqrt{\frac{1 - \cos \left(\theta\right)}{2}}} \pm 3 \sqrt{\frac{1 + \cos \left(\frac{\theta}{2}\right)}{2}}$

Step 4. Replace each $\theta \text{/} 2$ term with it its half-angle formula

$f \left(\theta\right) = \pm \frac{2}{\sqrt{\frac{1 \pm \sqrt{\frac{1 + \cos \left(\theta\right)}{2}}}{2}}} \pm \frac{\sqrt{\frac{1 \pm \sqrt{\frac{1 + \cos \left(\theta\right)}{2}}}{2}}}{\sqrt{\frac{1 - \cos \left(\theta\right)}{2}}} \pm 3 \sqrt{\frac{1 \pm \sqrt{\frac{1 + \cos \left(\theta\right)}{2}}}{2}}$

I'll leave simplifying this as homework!